2 D. Taniguchi et al. genitor (e.g. Fraser et al. 2011; Smartt 2015). Moreover, the Chiavassa et al.(2011b) and Davies et al.(2013) found that accuracy of Teff directly affects that of chemical abundances the TiO-band strength depends not only on Teff and the of RSGs, which could trace the abundance distribution of luminosity but also on the temperature structure, which is young stars in the Milky Way and nearby galaxies, leading significantly affected by granulation. to the galactic evolution theory (Patrick et al. 2017; Alonso- A promising way to circumvent the problems raised so Santiago et al. 2019; Origlia et al. 2019). Hence importance far is to use signals like weak atomic lines from the pho- of accurately determining Teff of RSGs with observations tosphere defined by the optical continuum or its outside cannot be overstated. close vicinity, given that they are expected to be less af- However, the accuracy of the reported Teff of RSGs is fected by some of the above-mentioned factors (Davies et al. still under debate. Interferometry and lunar occultation are 2013; Tabernero et al. 2018). A few works have presented the two simplest ways to measure Teff of nearby stars, but Teff of RSGs estimated on the basis of photospheric signals, these methods are subject to uncertainties in some or all of e.g., the SED of wavelengths less contaminated by molecu- the following three issues: (1) limb darkening (Dyck & Nord- lar lines (the SED method; Davies et al. 2013), equivalent gren 2002; Chiavassa et al. 2009), (2) the complex extended widths of some strong atomic lines in the J band (the J - envelope called MOLsphere (Tsuji 2006; Montarg`eset al. band technique; Davies et al. 2015) and some atomic lines 2014) and (3) interstellar reddening (Massey et al. 2005; around the Calcium triplet (Tabernero et al. 2018). In these Walmswell & Eldridge 2012). A more recent and improved three works Teff of common RSGs in the Magellanic Clouds way is to measure the stellar radius at continuum wave- were derived. Nevertheless some systematic offsets are ap- lengths to circumvent the effects of the MOLsphere with the parent between their results, possibly due to chemical abun- spectro-interferometric technique (e.g. Ohnaka et al. 2013; dances, contamination from some molecular lines to the sig- Arroyo-Torres et al. 2013); however, the result is still affected nals and/or some effect of strong lines which is in part by limb darkening and possibly uncertainty in interstellar formed in upper layers of the atmosphere. Moreover, the reddening unless it is negligible. departure from the 1D LTE condition may be important Another type of method to estimate Teff of the RSG in RSGs (e.g. Bergemann et al. 2012b; Kravchenko et al. is to model-fit molecular bands in relatively low-resolution 2019). Most of the above works assumed the 1D LTE, and spectra in the optical (e.g. Scargle & Strecker 1979; Oestre- this shortcoming could contribute to the offsets. icher & Schmidt-Kaler 1998). The reliability of this ap- Considering these discrepancies and the aforementioned proach has been improved thanks to the advancements of factors that possibly add some systematic error to the tem- the model of the stellar atmosphere such as, most notably, peratures estimated in most of the past methods, we here the MARCS model with improved handling of molecular take a different method that satisfies the following three con- blanketing (Gustafsson et al. 2008). The most comprehen- ditions. First, the method should use only relatively weak sive work of this approach was presented by Levesque et al. atomic lines that are not severely contaminated by molecu- (2005) for RSGs in the Milky Way, in which they made use of lar lines. Such lines do not originate in atmospheric layers far the optical TiO bands (see also Levesque et al. 2006; Massey above the photosphere, where the temperature structure is et al. 2009; Massey & Evans 2016, where RSGs in other not well constrained and moreover is expected to show vari- galaxies were studied with the same method); their method ability. Second, the method should be independent of stellar is hereafter referred to as the TiO method. parameters, abundances and reddening of both interstellar A similar approach is to compare photometric colours and circumstellar origins as much as possible. Third, ideally, of RSGs with the prediction of the MARCS model (e.g. the method should be independent of uncertainties in theo- Levesque et al. 2005; Drout et al. 2012; Neugent et al. 2012), retical models, in particular with regard to the factors orig- in which molecular lines are taken into account to calculate inating from the parameters in the upper atmosphere lay- the spectral energy distribution (SED) of the RSG. This ers. Therefore, we use the ‘line-depth ratio’ (LDR) method, approach tends to give a lower Teff in literature than the which relies on the empirical calibration of the relations be- TiO method by up to ∼ 200 K (Levesque et al. 2005, 2006), tween the LDR and Teff and satisfies the above-mentioned and hence there remain systematic uncertainties in fitting three requirements. In the next subsection, we explain what molecular absorption in optical spectra (Davies et al. 2013). the LDR and LDR-method are and review relevant past Alternatively some authors estimated Teff and spec- studies. tral types of RSGs from molecular CO, H2O, OH and/or CN lines in near-infrared spectra (e.g. Carr et al. 2000; 1.2 Line-depth ratio as an indicator of the effective Cunha et al. 2007; Origlia et al. 2019). However, Lan¸con temperature et al.(2007) and Lan¸con & Hauschildt(2010) found that the strengths of the CN molecular bands and the ratios of In cool stars (with a temperature of the solar tempera- the strengths of the CO bandheads could not be reproduced ture or lower), the depths of low-excitation lines of neu- with modern models of the static photosphere for the CNO tral atoms tend to be sensitive to Teff whereas those of abundances as predicted by stellar evolution models and a high-excitation lines are relatively insensitive (Gray 2008a). −1 microturbulent velocity (vmicro) of 2 km s . Therefore, the ratios of the depths between the low- and The temperature estimated on the basis of molecular high-excitation lines (that is, the LDRs) are good tempera- bands are affected more or less by a few factors, including ture indicators (Gray & Johanson 1991; Teixeira et al. 2016; chemical abundances and discrepancy between the atmo- L´opez-Valdivia et al. 2019, and references therein). The basic sphere of the real star and that estimated with a simpli- procedure is as follows: (1) search high-resolution spectra for fied static model assuming one-dimensional local thermody- the line pairs whose depth ratios are sensitive to Teff , (2) es- namic equilibrium (LTE) without MOLsphere. For example, tablish the empirical relations between LDRs and Teff for

MNRAS 000,1–19 (2020) Teff of RSGs using LDRs of Fe i lines in YJ bands 3 stars with well-determined Teff and (3) apply the relations mine their Teff in an unprecedented accuracy as non-spectro- to target objects to determine their Teff . interferometric measurements (Section 4). We focus on the near-infrared Y and J bands. These bands host a relatively small number of molecular lines in the RSG spectra (Coelho et al. 2005; Davies et al. 2010), and we can find sufficiently many atomic lines that are not 2 OBSERVATIONS AND REDUCTION contaminated by molecular lines (see Section 3.1 for detail 2.1 Sample and real examples). Taniguchi et al.(2018, hereafter Paper I) recently investigated LDRs in the Y and J bands of ten red We consider two groups of targets for this work. The first giants with well-determined effective temperatures (3700 < group consists of nine well-known red giants and are used for Teff < 5400 K). They calibrated 81 LDR–Teff relations with calibration of the LDR–Teff relations in this work. Five of the neutral atomic lines of various elements, and found that these stars (ε Leo, Pollux, µ Leo, Aldebaran and α Cet) were a precision of ±10 K is achievable in the best cases, i.e., selected from Gaia benchmark stars (Blanco-Cuaresma et al. high-resolution spectra of early M-type red giants with a 2014), whose Teff were well constrained with interferometry good signal-to-noise ratio (S/N). This precision rivals those by Heiter et al.(2015). The other four stars are well-studied achieved in the previous results in which the LDRs in optical nearby red giants selected from the MILES sample (S´anchez- high-resolution spectra were employed (e.g. Kovtyukh et al. Bl´azquez et al. 2006). The temperatures of these stars were 2006). determined using the ULySS program (Koleva et al. 2009) Although Paper I gave well-defined LDR–Teff relations by Prugniel et al.(2011) with the MILES empirical spec- for solar-metal red giants, some significant systematic un- tral library as a reference. The temperature scale of Prugniel certainty remains in their work, where a simple relation be- et al.(2011) relies on the compilation of literature stellar pa- tween the LDR and Teff was assumed to hold universally. In rameters, for which most spectroscopic works assumed the reality, LDRs depend on other stellar parameters than Teff , 1D LTE condition, and we have checked the consistency be- even though Teff is the dominant parameter that determines tween the temperature scale of Prugniel et al.(2011) and the LDR of a neutral atom. Jian et al.(2019), for example, that of Heiter et al.(2015) as follows. There are 13 stars 1 demonstrated that the LDR–Teff relations in the H band with [Fe/H] higher than −1.0 dex that were investigated in depend on the metallicity ([Fe/H]) by 100–800 K/dex. Also, both papers. The temperatures from the two studies differ Paper I suggested some effect of [Fe/H] and abundance ra- by 28 K on average with the unbiased standard deviation tios on their LDR–Teff relations. In this work, we limit the of the differences being 68 K, which is comparable with the sample to objects with [Fe/H] around solar (Section 2.1) to combined measurement errors. Thus, the temperature scale circumvent the problem of the [Fe/H] dependence. of Prugniel et al.(2011) is the same as that of Heiter et al. Surface gravity effect on LDRs has been also detected. (2015) within the errors, and we make use of the temper- Jian et al.(2020, hereafter Paper II) compared the LDRs of atures by Prugniel et al.(2011). The stellar parameters of the line pairs, reported by Paper I, between 20 dwarfs, 25 these nine red giants are summarized in Table 1. We also giants and 18 supergiants and reported the LDRs’ depen- adopted elemental abundances from Jofr´eet al.(2015) if dency on the surface gravity. This dependency is explained available. We note that these red giants were also used in with difference in the ionization stages among the elements Paper I to establish the LDR–Teff relations from pairs of involved in the line pairs. All the past attempts to derive lines with all possible combinations of the selected elements. a number of relations utilized any combination of usable The parameters Teff of the red giants range from 3700 K to species among selected elements; accordingly, their estimates 5400 K and are used in the calibration of the LDR–Teff re- might be significantly affected by the surface gravity. The lations (Section 3.2). The other stellar parameters are used LDRs of two lines with common species are, in contrast, in- only to check blending of absorption lines and to select the sensitive to the surface gravity. This kind of LDRs is also useful lines (Section 3.1). insensitive to the chemical abundance ratios. Therefore, the The second group of the targets consists of ten nearby LDR–Teff relations derived on the basis of line pairs of the RSGs and is the main target group in this work. Their [Fe/H] same species which are calibrated with the spectra of red gi- are close to solar (Luck & Bond 1989; McWilliam 1990; Wu ants are, when applied to RSGs, expected to yield Teff with- et al. 2011), as expected for young stars in the solar neigh- out introducing large systematic errors. Specifically, we use bourhood. The effective temperatures Teff of all these ten Fe i lines in this work because they are the only species that RSGs were estimated by Levesque et al.(2005) using the gives a sufficient number of the lines, hence a sufficient num- TiO method (3600 K < Teff < 4000 K); thus we can com- ber of the line pairs, with which well-constrained LDR–Teff pare our derived Teff values and theirs. Four of our sample, relations can be derived. Nevertheless, the difference in the NO Aur, V809 Cas, 41 Gem and ξ Cyg, were selected from size of the 3D non-LTE effect between red giants and RSGs the sample in Kovtyukh(2007) and they were used also by could introduce systematic errors to Teff of RSGs obtained Paper II except V809 Cas to investigate the LDR–Teff rela- with our method, and should be examined. tions of supergiants (3900 < Teff < 6300 K), where lines of In this work, using the high-resolution spectra in the various elements were employed. near-infrared Y and J bands of solar-metal red giants with well-determined stellar parameters observed with the WINERED spectrograph (Section 2), we construct a set 1 The 13 stars common in Prugniel et al.(2011) and Heiter of many empirical LDR–Teff relations with only Fe i lines et al.(2015) have HD numbers: 6582, 10700, 22049, 22879, 49933, for the first time (Section 3). Then, we apply these rela- 102870, 201091 (dwarfs), 18907, 23249, 121370 (subgiants), 29139, tions to the WINERED spectra of nearby RSGs and deter- 85503 and 220009 (giants).

MNRAS 000,1–19 (2020) 4 D. Taniguchi et al.

Table 1. The stellar parameters of the calibrating nine red giants. Full-width at half maximum (FWHM) indicates the broadening width assuming Gaussian function that includes instrumental (∼ 10.7 km s−1), macroturbulent and rotational broadenings.

−1 −1 [7] Name Teff [K] [Fe/H] [dex] log g [dex] vmicro [km s ] FWHM [km s ] ε Leo 5398 ± 31[2] −0.06 ± 0.04[2] 2.02 ± 0.08[2] 1.61[6] 15.91 κ Gem 5029 ± 47[2] −0.01 ± 0.05[2] 2.61 ± 0.12[2] 1.47[6] 12.49 ε Vir 4983 ± 61[1] +0.10 ± 0.16[4] 2.77 ± 0.02[1] 1.39 ± 0.25[5] 12.42 Pollux 4858 ± 60[1] +0.08 ± 0.16[4] 2.90 ± 0.08[1] 1.28 ± 0.21[5] 11.74 μ Leo 4470 ± 40[3] +0.20 ± 0.15[4] 2.51 ± 0.11[1] 1.28 ± 0.26[5] 12.56 Alphard 4171 ± 52[2] +0.08 ± 0.07[2] 1.56 ± 0.20[2] 1.66[6] 12.71 Aldebaran 3882 ± 19[3] −0.42 ± 0.17[4] 1.11 ± 0.19[1] 1.63 ± 0.30[5] 13.19 α Cet 3796 ± 65[1] −0.50 ± 0.47[4] 0.68 ± 0.23[1] 1.77 ± 0.40[5] 12.87 δ Oph 3783 ± 20[2] −0.03 ± 0.06[2] 1.45 ± 0.19[2] 1.66[6] 12.67 References: [1] Heiter et al.(2015); [2] Prugniel et al.(2011); [3] Weignted mean of values in Heiter et al.(2015) and Prugniel et al. (2011); [4] Heiter et al.(2015) converted to Solar metallicity of A(Fe) = 7.50 dex (Asplund et al. 2009); [5] Jofr´eet al.(2015); [6] Estimated using the relation between log g and vmicro for ASPCAP DR13 (Holtzman et al. 2018); [7] Measured by the fitting of several isolated lines.

Table 2. Observation log of our sample stars. Spectral types are taken from SIMBAD (Wenger et al. 2000) on 2020 April 26.

Object Telluric Standard Obs. Date Name HD Sp. Type Name HD Sp. Type Nine well-known red giants ε Leo 84441 G1IIIa 21 Lyn 58142 A0.5Vs 2014 Jan 23 κ Gem 62345 G8III–IIIb HR 1483 29573 A0V 2013 Dec 8 ε Vir 113226 G8III–IIIb b Vir 104181 A0V 2014 Jan 23 Pollux 62509 K0IIIb HIP 58001 103287 A0Ve+K2V 2013 Feb 28 μ Leo 85503 K2IIIbCN1Ca1 HIP 76267 139006 A1IV 2013 Feb 23 Alphard 81797 K3IIIa HR 1041 21402 A2V 2013 Nov 30 Aldebaran 29139 K5+III HIP 28360 40183 A1IV–Vp 2013 Feb 24 α Cet 18884 M1.5IIIa omi Aur 38104 A2VpCr 2013 Nov 30 δ Oph 146051 M0.5III b Vir 104181 A0V 2014 Jan 23 Ten nearby RSGs ζ Cep 210745 K1.5Ib HR 6432 156653 A1V 2015 Aug 8 41 Gem 52005 K3–Ib HR 922 19065 B9V 2015 Oct 28 ξ Cyg 200905 K4.5Ib–II 39 UMa 92728 A0III 2016 May 14 V809 Cas 219978 K4.5Ib c And 14212 A0V 2015 Oct 31 V424 Lac 216946 K5Ib HR 8962 222109 B8V 2015 Jul 30 ψ1 Aur 44537 K5–M1Iab–Ib HIP 53910 95418 A1IVps 2013 Feb 22 TV Gem 42475 M0–M1.5Iab 50 Cnc 74873 A1Vp 2016 Jan 19 BU Gem 42543 M1–M2Ia–Iab 50 Cnc 74873 A1Vp 2016 Jan 19 Betelgeuse 39801 M1–M2Ia–Iab HIP 27830 39357 A0V 2013 Feb 22 NO Aur 37536 M2Iab HR 922 19065 B9V 2015 Oct 28

2.2 Observations 52nd (Y band; 0.97–1.09 µm) and 48th–43rd (J band; 1.15–1.32 µm) only among the orders 61st–42nd because stel- All the objects were observed using the near-infrared high- lar atomic lines in the unselected orders are severely contam- resolution spectrograph WINERED installed on the Nas- inated by other lines. The orders 61st–59th, 50th–49th and myth platform of the 1.3 m Araki Telescope at Koyama 42nd are dominated by telluric lines in our spectra collected Astronomical Observatory of Kyoto Sangyo University in in Kyoto (see Figure 4 in Sameshima et al. 2018). The or- Japan (Ikeda et al. 2016). We used the WINERED WIDE ders 58th and 51st are dominated by stellar CN molecular mode to collect spectra covering a wavelength range from 0 lines of the (1–0) bandhead and those of the (0–0) band 0.90 to 1.35 μm (z , Y and J bands) with a spectral reso- of the CN A2Π–X2Σ+ red system, respectively (e.g. Ku- lution of R ∼ 28, 000. We selected the nodding pattern of rucz 2011; Brooke et al. 2014). These two orders are heavily A–B–B–A or O–S–O. All our targets are bright (−3.0 ≤ contaminated also by telluric lines in the spectra taken in J ≤ 3.0 mag), and the total integration time for each target summer. The wavelength ranges of the individual orders of within the slit ranged between 3–240 sec, with which a S/N WINERED are given in Table 4. per pixel of 100 or higher was achieved. Telluric standard stars (slow-rotating A0V stars in most cases; see Sameshima et al. 2018) were also observed, the spectra of which were 2.3 Data reduction used to subtract the telluric absorption. Table 2 summarizes the observation log. The initial steps of the spectral reduction were performed As in Paper I, we utilized the echelle orders 57th– with the WINERED data-reduction pipeline (Hamano et

MNRAS 000,1–19 (2020) Teff of RSGs using LDRs of Fe i lines in YJ bands 5 Sr II Y II Ni I Fe I Fe I Fe I Fe I ? Fe I Ca I ? Fe I Fe I Fe I Fe I Co I Fe I ? Fe I Fe I

HD 210745 (K1.5 Ib) 1.0

HD 52005 (K3- Ib) 0.7

HD 200905 (K4.5 Ib-II) 0.4

HD 219978 (K4.5 Ib) 0.1

HD 216946 (K5 Ib) 0.2

HD 44537 (K5-M1 Iab-Ib) 0.5

HD 42475 (M0-M1.5 Iab) 0.8 Normalized Flux + Offset HD 42543 (M1-M2 Ia-Iab) 1.1

HD 39801 (M1-M2 Ia-Iab) 1.4

HD 37536 (M2 Iab) 1.7

RSG3 Model 2.0

Atomic CN FeH

10325 10330 10335 10340 10345 10350 10355 10360 10365 Air Wavelength [Å]

Figure 1. A part of the reduced WINERED spectra and the model RSG3 spectrum in the echelle order 54th with some identified lines labelled. Four Fe i lines indicated by red arrows are used for the final set of the LDR line pairs (Table 6). We have not identified the origins of the three stellar lines labelled with ‘?’, among which the line at 10338.5 A˚ is catalogued in the list of unidentified lines by Matsunaga et al.(2020).

MNRAS 000,1–19 (2020) 6 D. Taniguchi et al.

Table 3. Stellar parameters used to simulate spectra of red giants (3) systematic errors caused by slightly wrong continuum and RSGs; we use these spectra for various purposes (see text). placement. Though the errors introduced by the first source We assumed the solar chemical abundance pattern (Asplund et al. can be estimated using a certain method, e.g., the one by 2009). Fukue et al.(2015, see their Section 3.2), those by the sub- sequent sources cannot be estimated in a straightforward Model RSG1 RSG2 RSG3 RGB1 manner. Sp. Type K3I M1.5I M0I M0.5III In order to estimate realistic errors in line depths in- Teff [K] 4000 3700 3850 3850 cluding all the above-mentioned three sources, we consid- [Fe/H] [dex] 0.0 0.0 0.0 0.0 ered the ‘effective’ S/N per pixel of the telluric-subtracted log g [dex] 0.5 0.0 0.25 1.0 spectra as follows. For this, we used the ready-to-use tar- −1 vmicro [km s ] 1.8 2.5 2.15 1.7 get spectra with normalized continuum. First, we measured FWHM [km s−1] 16 21 18.5 13 the deviations, δi, from unity of each pixel, i, within the continuum regions selected in Section 2.3.1. The pixels {i} al. in preparation). Some details of the pipeline are given in that satisfy |δi| > 0.05 might not be in real continuum re- Section 3.1 in Paper I. Then, the telluric absorption lines gions, and thus were excluded in the subsequent analysis. Of were removed using the observed spectra of A0V stars af- course, such a threshold cannot be used for low-S/N spectra, ter their intrinsic lines had been removed with the method but our spectra are sufficiently high ‘effective’ S/N, ∼ 50 at described in Sameshima et al.(2018), except for the 55th– least. We note that some of such pixels might be due to the 53rd orders (1.01 to 1.07 μm) of the objects taken in winter, absorption lines in the observed RSG spectra that were not in which almost no significant telluric lines were present. The predicted in the synthesized spectra, and others might be in- radial velocities were measured and corrected by comparing duced by the bad continuum normalization. The ‘effective’ the observed and synthesized spectra. Finally the contin- S/N were then estimated according to, after three iterations uum was re-normalized and the realistic S/N was estimated of three-sigma clipping, as we describe in the following subsections. An example of −1/2  P δ 2  the reduced spectra of RSGs is presented in Figure 1. S/N = i i , (1) Npixel − 1

where Npixel is the number of the used pixels. We note that 2.3.1 Continuum Normalization the sigma-clipping may underestimate the error, but by only The nominal continuum normalization was automatically 2% at most, providing that δi follows the Gaussian distri- performed in the data-reduction pipeline, but the automatic bution. The resultant S/N of the RSGs are summarized in normalization often yields unsatisfactory results, especially Table 4, whereas those for the red giants are found in Ta- for line-rich stars like RSGs. Therefore, we further optimized ble 2 in Paper I. The reciprocal of the S/N measured here the normalizations for our spectra in the following proce- are henceforth regarded as the error in line depth. We note dure. First, we prepared the target spectra with the telluric that the ‘effective’ S/N of RSGs could be under- or overes- absorption lines subtracted and with the wavelength-scale timated to some extent because, for example, weak stellar adjusted to the one in the air at rest. Second, we selected absorption lines that are not recognized in the chosen contin- a set of continuum regions for each group of the red giants uum regions may exist in reality. Moreover, the wavelength and RSGs, where the spectra are not significantly affected ranges in which many absorption features exist may be af- by the stellar lines. For the nine red giants, the continuum fected by the normalization error more than indicated by the regions were selected with visual inspection. In contrast, we ‘effective’ S/N because it is harder to trace the continuum at searched for the continuum regions of the RSGs by choosing these regions than the selected regions without absorption the wavelength ranges with the depths from unity smaller lines. than 0.3% in the model spectrum with the stellar parame- ters of ‘RSG3’ in Table 3 (see Section 3 about the spectral synthesis). These steps left a few tens of evenly distributing 3 CALIBRATION OF THE TEMPERATURE SCALE continuum segments in each order. Finally, we normalized USING RED GIANTS the continuum of each order of each target star with the in- teractive mode of the IRAF continuum task, in which we In this section, we establish the key relations of this work, mainly used the selected continuum regions but sometimes i.e., a set of the reliable empirical relations between the LDR further combining some continuum regions visually selected and Teff , in the following strategy. First, we choose Fe i lines for each order of each star. that are relatively free from blending by other lines. Then we find the best pairs whose LDRs show a well-defined cor- relation with Teff . 2.3.2 Signal-to-Noise Ratios of Telluric-Subtracted Spectra For the spectral synthesis employed in this section, we developed a wrapper software in Python3 of the spectral We need to know the errors in line depth for calibrating synthesis code MOOG (Sneden 1973; Sneden et al. 2012)2. the LDR–Teff relations and for estimating Teff . The depth MOOG synthesizes spectra, assuming the 1D LTE with the errors are considered to originate in three sources: (1) sta- plane-parallel geometry. We remark that this assumption tistical noises in the telluric-subtracted spectra, which can be estimated with a combination of S/N of the target and telluric standard spectra (200 per pixel or higher in most 2 We used the February-2017 version of MOOG further modified of the cases), (2) residuals in the telluric subtraction and by M. Jian (https://github.com/MingjieJian/moog_nosm).

MNRAS 000,1–19 (2020) Teff of RSGs using LDRs of Fe i lines in YJ bands 7

Table 4. ‘Effective’ S/N per pixel of the reduced spectra of RSGs in each echelle order (57th to 52nd in the Y band and 48th to 43rd in J) and the wavelength coverage, λmin < λair < λmax.

Object m57 m56 m55 m54 m53 m52 m48 m47 m46 m45 m44 m43 ζ Cep 75 112 70 86 108 113 57 56 68 120 80 73 41 Gem 55 149 123 119 107 117 50 61 74 147 63 81 ξ Cyg 98 151 92 148 138 84 60 55 106 138 93 61 V809 Cas 103 122 58 100 127 91 60 64 95 128 99 65 V424 Lac 57 59 68 99 80 69 54 57 83 147 58 66 ψ1 Aur 109 138 105 144 71 64 86 135 75 167 81 103 TV Gem 89 98 107 105 82 68 66 58 78 191 51 57 BU Gem 73 69 77 75 81 73 85 53 57 109 70 54 Betelgeuse 49 53 55 87 117 75 89 89 60 146 64 60 NO Aur 66 150 59 85 133 89 46 54 70 123 85 72

λmin [µm] 0.976 0.993 1.011 1.029 1.049 1.069 1.156 1.181 1.206 1.233 1.260 1.289 λmax [µm] 0.993 1.011 1.029 1.049 1.069 1.089 1.181 1.206 1.233 1.260 1.289 1.319

does not affect our final determination of Teff , which relies in RSGs (Tsuji 2000; Kervella et al. 2009). Therefore, we only on the empirical LDR–Teff relations, though the selec- selected the Fe i lines that are sufficiently free from blend- tion of the candidate lines could be affected to some extent. ing with lines of molecules and atoms other than Fe i it- The grid of the model atmospheres of RSGs and red gi- self. Moreover, contamination by other Fe i lines may also ants was taken from the MARCS (Gustafsson et al. 2008) introduce a gravity effect, given that the LDRs of the ob- with the spherical geometry (M = 2M for most cases) served depths of blended lines are affected by microturbulent assumed, and the model atmosphere at a set of finer-scale and macroturbulent velocities (Biazzo et al. 2007), both of stellar parameters was obtained with linear interpolation. which tend to be larger at lower surface gravities (Holtzman We note that although the geometry of the radiative trans- et al. 2018). Therefore, we also checked the contamination fer code (plane parallel) is different from that of the model by other Fe i lines. atmospheres assumed here (spherical), the difference in their In order to evaluate the effect of the potential blending, geometries is known to induce only a small effect in gen- we used synthetic spectra of the two model RSGs (RSG1 eral on the synthesized spectra (Heiter & Eriksson 2006). and RSG2 in Table 3) and those of the five coolest red gi- As for the line list, we used the third release of the Vienna ants (µ Leo, Alphard, Aldebaran, α Cet and δ Oph). The Atomic Line Database (VALD3; Ryabchikova et al. 2015) as stellar parameters of RSG1 and RSG2 roughly correspond the main source. The molecular lines in the Y and J bands to those of the warmest and coolest RSGs in our sample, re- contained in the VALD3 are limited to CH, OH, SiH, C , 2 spectively, and the line blendings in RSG2 are expected be CN and CO, among which only CN gives a significant ab- severer than those in all the sample observed RSGs. With sorption in the spectra of our target RSGs and red giants. In each set of stellar parameters we measured four types of addition, we adopted the line list of FeH by B. Plez (private depths (d , d , d and d ) for each Fe i communication)3. As a result of the extensive examination, All OneOut SameElIonOut AtomOut line in VALD3 (let us use λ to denote the centre wave- we found that lines of FeH appeared in the spectra of the 0 length listed in VALD3) as follows. First, we synthesized RSGs with a depth up to ∼ 10% and that those lines were the respective four types of spectra around the target line well reproduced by synthesized spectra with a dissociation with different groups of the lines included: (1) All—all the energy of 1.59 eV (Schultz & Armentrout 1991). atomic and molecular lines, (2) OneOut—all the lines ex- cept for the target Fe i line, (3) SameElIonOut—all the lines 3.1 Line selection except any Fe i lines and (4) AtomOut—only the molecular lines (see an example in Figure 2). Next, we identified the We chose the Fe i lines that are sufficiently strong and rel- absorption around λ0 in the synthesized spectrum All and atively free from blending out of the list of the Fe i lines determined the peak wavelength, λc, the value of which is in the VALD3. Since our aim is to estimate Teff of RSGs different from λ0 if the target line is blended, or identical to using the LDR-Teff relations calibrated with red giants in it if not. Finally, the respective depths from unity at λc in spite of the difference in the surface gravity between the two all the four types of the synthesized spectra were measured. groups, it is necessary to choose the Fe i lines that are least The reason why we measured the depths at λc rather than sensitive to the surface gravity. The LDR of a pair of lines λ0 in the synthesized spectra is that we measured the depth from two different elements, especially those with different of each line in the observed spectra at the wavelength of the ionization energies, is known to show dependence on the sur- peak position around the line rather than the wavelength in face gravity (Paper II). Also, depths of molecular lines are the line list λ0 (Section 3.2). significantly different between RSGs and red giants because of their strong dependence on the surface gravity (Lan¸con With these depths, we considered two types of criteria et al. 2007), non-solar CNO abundances (Lambert et al. for selecting lines. First, we imposed dAll > 0.02 on the 1984; Ekstr¨om et al. 2012) and existence of MOLspheres synthetic spectra of RSG1. When two or more lines were assigned to the same wavelength of minimum λc in RSG1, we would consider only the line that contributes most to the 3 https://www.lupm.in2p3.fr/users/plez/ peak, i.e. the line with the smallest dOneOut, as a candidate.

MNRAS 000,1–19 (2020) 8 D. Taniguchi et al.

thesized spectra of the four coolest red giants and this line Fe I, 0 = 10742.550Å meets the criteria, but the observed depths are too small, 1.00 0.010–0.019. Fourth, we excluded two lines, Fe i λ10780.694 molecule and λ12340.481 A,˚ because they were affected with diffuse 0.98 molecule interstellar bands λ10780 and λ12337, respectively, reported + 0.96 other atoms by Hamano et al.(2015). Fifth, hydrogen Paschen series and + helium 10830 A˚ lines are known to show large variations be- 0.94 other FeI tween stars, especially in supergiant stars (e.g. Obrien & 0.92 Lambert 1986; Huang et al. 2012), and hence may affect all the blending significantly. However, we found that none of Normalized Flux 0.90 molecule the selected lines were affected by them in our case. Sixth, + Matsunaga et al.(2020) detected dozens of lines that are not 0.88 other atoms found in the available line lists in the YJ -band spectra of su- 0.86 c pergiants (4000 < Teff < 7200 K), and these lines may affect 10741 10742 10743 10744 0 our result significantly. However, all the lines selected here Air Wavelength [ ] Å were separated by more than 30 km s−1 from these uncata- logued lines (Table 3 of Matsunaga et al. 2020), and thus we safely ignored this potential influence. Figure 2. Examples of the synthesized spectra of RSG3 around Fe i Applying all these points to the candidate lines, we ob- ˚ λ10742.550 A(λ0) with different lines included. Cyan, orange, red tained in the end 52 Fe i lines in total (28 in the Y band and pink lines show the spectra of All, OneOut, SameElIonOut and 24 in J ) and summarize the result in Table 5. We note and AtomOut, respectively, defined in the main text. The peak that Paper I considered another criterion requiring that the wavelength in All (λ ) was indicated with the vertical orange c depth of each line be smaller than 0.5 in the observed spec- dashed line. We note that this line in this figure shows one of the most complex blended lines among the final set of the selected tra of Arcturus; this condition is satisfied in all the lines that lines. we selected. Then, we measured the depths of the selected 52 Fe i lines in each observed spectrum by fitting a quadratic func- Then, we imposed the following three criteria, which should tion to three (or four) pixels centred at the peak of each be satisfied in all the seven synthetic spectra of the RSGs (n) line (Strassmeier & Schordan 2000). Here we define d as and the red giants: i the line depth from the continuum level to the peak of the  fitted function, where i denotes the ID number of the line dOneOut/dAll < 0.5  and (n) does the ID number of the star. We ignored the dSameElIonOut/dAll < 0.3 . (2) measurements if d(n) < 0.02, if the continuum around the d /d < 0.2 i AtomOut All line was not well normalized (only λ9889.0351 A˚ in Alde- Applying the combination of all the above-mentioned crite- baran) or if the measured centre wavelength λc was not −1 ria to the current line list left 76 Fe i lines in total (41 in the within ±10 km s of the corresponding wavelength λ0 in Y band and 35 in J ). We note that the index dOneOut/dAll the VALD3 line list. We considered the reciprocal of the ‘ef- had been used in evaluating line blending in some of our fective’ S/N per pixel estimated in Section 2.3.2 to be the recent papers (Kondo et al. 2019; Matsunaga et al. 2020; (n) error in di , which includes both the systematic errors (e.g., Paper II); the difference in this work was that two slightly in the telluric subtraction and the continuum normalization) different indices, dSameElIonOut and dAtomOut, were added to the and the statistical errors (Poisson, read-out and background condition to give tighter constraints. noises). Furthermore, we considered the following six points, with which some candidate lines would be filtered out. First, some of the observed lines were found to be con- 3.2 Line-pair selection: method taminated by unexpected stellar lines. For example, though Fe i λ12393.067 A˚ is expected to be blended only with Fe i We searched for the best set of line pairs of which the LDR– λ12393.281 A˚ in the synthetic spectrum and to meet the Teff relations yield the most precise estimates of Teff . Here, criteria described above, the peak wavelengths λc in the ob- we followed the procedures in Paper I and Paper II except served spectra of all the red giants were found to be sys- that we combined the lists of Fe i lines in different echelle tematically shorter by ∼ 0.25 ˚A than the wavelength (λ0) orders in each of the Y and J bands. The basic idea of listed in the VALD3 line list; this fact implies that this the process is to select the set of line pairs that meet the line is blended with an unknown line(s) with its peak wave- following conditions best: (1) high precision in reproducing length at around 12392.7 A.˚ Second, some of the observed Teff of our sample stars and (2) small difference in wave- lines were contaminated by residuals from telluric subtrac- length between the two lines of each line pair. The latter tion or suffered from imperfect continuum normalization due condition is desirable in the sense that different instruments to many stellar lines, especially CN molecular lines, concen- from WINERED in future observations would have a better trated in the close vicinity of the line. Third, some of the chance to be capable of detecting both lines of the line pair observed lines were found to be much weaker than those simultaneously. in the synthetic ones possibly due to the inaccurate oscil- First, we evaluated the relation between LDR and Teff of lator strength values in the VALD3 line list. For example, all the possible line pairs individually. In this case, 211 and the depths of Fe i λ10462.155 A˚ were 0.016–0.028 in the syn- 128 pairs in the Y and J bands, respectively, were found

MNRAS 000,1–19 (2020) Teff of RSGs using LDRs of Fe i lines in YJ bands 9

Table 5. List of the 52 selected Fe i lines. The last column is defined Total Least Squares method (see Markovsky & Van Huffel when the line is used in the final set of the LDR–Teff relations, 2007, for a review). For this regression, the weight of each and shows the ID number of the LDR. point was given by

 2 2−1 Lower-level Upper-level log gf LDR (n)  (n) 2  (n) wj = ∆Teff + aj σj , (3) λair [A]˚ E [eV] Term E [eV] Term [dex] ID 9800.3075 5.0856 x5F◦ 6.3503 e5G −0.453 where ∆T (n) and σ(n) indicate the standard errors of T (n) 9811.5041 5.0117 y7P◦ 6.2750 e7P −1.362 eff j eff 5 ◦ 7 (n) 9868.1857 5.0856 x F 6.3416 e F −0.979 (1) in literature and of log rj for each star, respectively. The 9889.0351 5.0331 x5F◦ 6.2865 e5G −0.446 weighted dispersion with regard to each regression line was 10065.045 4.8349 y3D◦ 6.0663 e3F −0.289 (2) then calculated as 3 5 ◦ 10081.393 2.4242 a P2 3.6537 z P −4.537 (1) v 5 5 ◦ u N 10155.162 2.1759 a P 3.3965 z F −4.226 h i2 3 ◦ 3 u X (n) (n) (n) 10216.313 4.7331 y D 5.9464 e F −0.063 (3) u wj Teff − (aj log rj + bj ) 3 3 ◦ u 10218.408 3.0713 c P 4.2843 z P −2.760 u N n=1 σj = u . (4) 10265.217 2.2227 a5P 3.4302 z5F◦ −4.537 (4) u N − 2 N 3 3 ◦ u X (n) 10332.327 3.6352 b D 4.8349 y D −2.938 (5) t wj 5 5 ◦ 10340.885 2.1979 a P 3.3965 z F −3.577 n=1 10347.965 5.3933 w5D◦ 6.5911 f 5P −0.551 (6) 10353.804 5.3933 w5D◦ 6.5904 h5D −0.819 (7) We set the threshold of σj < 150 K for the pair to be valid. 10364.062 5.4457 w5D◦ 6.6417 f 5P −0.960 (5) As a result, 41 and 6 line pairs in the Y and J bands, re- 10395.794 2.1759 a5P 3.3683 z5F◦ −3.393 (8) spectively, were left in the subsequent analyses. We also set 3 3 ◦ 10423.027 2.6924 a G 3.8816 z F −3.616 (6) another threshold aj > 0, but no line pairs were rejected 10423.743 3.0713 c3P 4.2605 z3P◦ −2.918 (3) with this condition. 3 ◦ 3 10532.234 3.9286 z D 5.1055 P −1.480 (8) Second, we searched for the optimal set of the line pairs 5 ◦ 5 10555.649 5.4457 w D 6.6200 g F −1.108 (9) for each of the Y and J bands as follows. Each set of line 10616.721 3.2671 b3H 4.4346 z3G◦ −3.127 (4) pairs can be considered as an undirected graph whose nodes 10725.185 3.6398 b3D 4.7955 y3D◦ −2.763 (7) 10742.550 3.6416 b3D 4.7955 y3D◦ −3.629 (2) correspond to absorption lines and whose edges connecting 10753.004 3.9597 z3D◦ 5.1124 3P −1.845 the nodes correspond to line pairs. We require that each line 10754.753 2.8316 b3P 3.9841 z3F◦ −4.523 (9) be used only once in each set, i.e. no duplication of a line 10783.050 3.1110 c3P 4.2605 z3P◦ −2.567 in separate pairs are allowed. This ensures that the errors 3 ◦ 3 (n) 10818.274 3.9597 z D 5.1055 P −1.948 (10) in rj values in different LDR–Teff relation be independent 10849.465 5.5392 e5D 6.6817 5D◦ −1.444 (10) of each other and thus makes it straightforward to calcu- 5 5 ◦ 11638.260 2.1759 a P 3.2410 z D −2.214 late the statistical errors of the combined Teff (Equation 8 1 3 ◦ 11681.594 3.5465 a D 4.6076 y F −3.615 and Equation 9). We determined the optimal matching, M, 3 3 ◦ 11783.265 2.8316 b P 3.8835 z D −1.574 that meets our requirements as follows. We considered the 11882.844 2.1979 a5P 3.2410 z5D◦ −1.668 maximum matching, where the number of the edges of the 11884.083 2.2227 a5P 3.2657 z5D◦ −2.083 12119.494 4.5931 d3F 5.6158 y3G◦ −1.635 matching is as large as possible under the condition where no 12190.098 3.6352 b3D 4.6520 y3F◦ −2.330 nodes are connected with more than one edge. In the ideal 12213.336 4.6382 y5P◦ 5.6531 e5D −1.845 case, the size of the maximum matching, |M|, corresponds to 12267.888 3.2740 a3D 4.2843 z3P◦ −4.368 (11) a half of the number of all the selected lines. However, it did 3 5 ◦ 12556.996 2.2786 a P2 3.2657 z D −3.626 (12) not in our case because many edges were rejected due to, for 5 ◦ 5 12615.928 4.6382 y P 5.6207 e D −1.517 (11) example, large scatters around the corresponding LDR–Teff 5 ◦ 5 12638.703 4.5585 y P 5.5392 e D −0.783 relations. For a given maximum matching, Mk, of this undi- 5 ◦ 5 12648.741 4.6070 y P 5.5869 e D −1.140 (12) rected graph, we calculated the weighted mean temperature 12789.450 5.0095 x5D◦ 5.9787 e5F −1.514 (n) T on the basis of the LDR–Teff relations for each star. We 12807.152 3.6398 b3D 4.6076 y3F◦ −2.452 Mk also considered the difference in wavelength between the two 12808.243 4.9880 x5D◦ 5.9558 e5F −1.362 12824.859 3.0176 b3G 3.9841 z3F◦ −3.835 lines of each line pair jk,m, denoted as ∆λm,k, and calculated 3 5 ◦ the evaluation function E(Mk; e) defined as 12879.766 2.2786 a P2 3.2410 z D −3.458 12896.118 4.9130 x5D◦ 5.8741 e5F −1.424 v v N |M | 5 ◦ 7 u 2 u k 12934.666 5.3933 w D 6.3515 e G −0.948 u 1 X  (n) (n) u 1 X 2 13006.684 2.9904 b3G 3.9433 z3F◦ −3.744 E(Mk; e) = t TM − Teff + et (∆λk,m) N k |Mk| 13014.841 5.4457 w5D◦ 6.3981 f 5F −1.693 n=1 m=1 13098.876 5.0095 x5D◦ 5.9558 e5F −1.290 (5) 13147.920 5.3933 w5D◦ 6.3360 f 5F −0.814 = ET (Mk) + eEλ(Mk),

where ET (Mk) represents the size of the error in redeter- to have two lines that were both detected in more than four mining Teff of the nine stars for a given matching Mk, and stars and have excitation potentials (EPs) separated by more Eλ(Mk) represents the wavelength difference of the line pairs than 1 eV. For each pair, denoted with the subscript j, of and works as a penalty term. There are different allowed all the 211 + 128 line pairs, we calculated the LDR denoted combinations of the line pairs which form different maximum (n) as rj and its error. We plotted Teff against the common matchings, and for a given e value we selected the one that logarithms of the LDRs (log rj ) for each pair and determined gives the least E(Mk; e) as the optimal matching, Mk(e). the regression line Teff = aj log rj + bj , using the Weighted We would determine the coefficient e by considering how

MNRAS 000,1–19 (2020) 10 D. Taniguchi et al.

ET (Mk(e); e) and Eλ(Mk(e); e) depend on e, as explained in the following subsection. 140 Taniguchi+18 (each order) This work (Y band) 120 This work (J band) 3.3 Line-pair selection: results This work (all the bands) 100 Taniguchi+18 (all the orders)

We applied the procedure described in the previous sub- ] Fukue+15 (H band) K 80 [ section to our 28 and 24 nodes (i.e. lines) and 41 and 6 T edges (i.e. line pairs) in the Y and J bands, respectively. E 60 We searched for the optimized matching, Mk(e), that gives the smallest E at each e value between 0 and 2 K/A˚ to see 40 how e would affect the solutions. Figure 3 plots the values of 20 ET and Eλ for Mk(e) with varying e. Larger the parameter e is, larger the weight on Eλ is relative to ET in the total eval- 0 uation function E, and then the optimal matching and the 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 values of ET and Eλ change. We decided to employ the same e [K/Å] value, e = 0.5 K/A,˚ as Paper I because ET at e = 0.5 K/A˚ 500 is similar to that at e = 0 K/A,˚ which optimizes the preci- sion in the redetermined temperature, whereas Eλ is suffi- ciently small. Moreover, the same matching would have been 400 selected if another value of e among a wide range of e val- ues (0.39 < e < 1.81 K/A)˚ was employed. 300

Consequently, we obtained 10 and 2 line pairs in the Y ] Å [ and J bands, respectively, i.e., 12 line pairs in total (Table 6 E and Figure 4). This set of line pairs made use of 24 unique 200 lines in the Y and J bands combined (nb., no duplications of the lines are allowed; see Section 3.2). This condition has only a weak effect on the statistical error, which could have 100 been better by only ∼ 10% even if we had used all the 41+6 pre-selected pairs. The number of the selected lines (52 in 0 total, or 28 in Y and 24 in J ; see Section 3.1) is about one- 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 fourth of that used by Paper I (224 in total, or 125 in Y and e [K/Å] 99 in J ), where atomic lines of various elements were used. Accordingly, the total number of the final line-pairs, 12, is smaller than 81 in Paper I. Nevertheless, the number 12 is Figure 3. Evaluation functions, ET and Eλ, of the coefficient e (see larger than the number, 8, of the Fe i line pairs in Paper I. text). Blue and green thick lines show those for the Y and J ˚ The evaluation functions with e = 0.5 K/A with all the bands, respectively, in this work. Black thin lines show those of all line pairs in the Y and J bands combined are ET = 43.9 K 57th–52nd and 48th–43rd orders by Paper I. Red filled circles, all ˚ all and Eλ = 289.3 A. The former ET in this work is only magenta open squares and cyan open triangles show the evalua- 1.5 times larger than the counterpart in Paper I despite the tion functions of all the orders (or bands) combined, where the smaller number of the line pairs and smaller depths, both of line pairs selected in this work, Paper I and Fukue et al.(2015), which could have increased the statistical errors in T (n). The respectively, are employed. Mk all relatively small difference in ET indicates that the errors in all literature Teff (19–65 K) mainly contribute to ET in Paper I. all In contrast, the latter Eλ in this work is 4.3 times larger than that by Paper I and is similar to that by Fukue et al. (2015) for the H band. This is expected because Fukue et al. S S  where aa ab is the variance-covariance matrix of the (2015) and we treated lines in each band together, whereas Sab Sbb Paper I treated those in each echelle order independently. coefficients (aj , bj ) of the regression line. The value referred These differences in ET and Eλ are well demonstrated in to as the LDR effective temperature TLDR of the target Figure 3. star is defined as the weighted mean of the temperatures estimated from the available line pairs among the 12 line pairs (Tj ± ∆Tj ; for j = 1, ··· ,Npair). We define two forms 3.4 Re-determination of the effective temperatures of red of the error for TLDR, given by the following two formulae, giants and calculated both of these for each target, With each line pair j, the effective temperature and its error N 1/2  pair 2  of a target star (RSG in our case) would be estimated to be X (Tj − TLDR)  ∆T 2  T = a log r + b , (6)  1 j=1 j  j j j j ∆T Eq8 =   (8) LDR  N  p 2 2  Npair − 1 pair  ∆Tj = (aj ∆ log rj ) + Saa(log rj ) + 2Sab log rj + Sbb,  X 1  (7) ∆T 2 j=1 j

MNRAS 000,1–19 (2020) Teff of RSGs using LDRs of Fe i lines in YJ bands 11

Table 6. List of low- and high-excitation Fe i lines and the LDR–Teff relations; a and b denote the coefficients in Teff = a log r + b, the three S values (Saa, Sab and Sbb) represent the variance-covariance matrix of a and b, N is the number of stars used in the fitting, and σ is the weighted dispersion given by Equation 4. The line pair (6) in this table is included in Table 4 of Paper I with the ID (22).

Low-excitation line High-excitation line LDR–Teff relation ˚ ˚ ID Band Order λair [A] EP Order λair [A] EP a b Saa Sab Sbb N σ [eV] [eV] [K] [K] [103K2] [102K2] [102K2] [K] (1) Y m56 10081.393 2.424 m57 9868.1857 5.086 −2408 3976 37.5 26.0 15.5 9 133 (2) Y m52 10742.550 3.642 m56 10065.045 4.835 −3891 1149 3803.8 26919.2 19154.3 5 131 (3) Y m54 10423.743 3.071 m55 10216.313 4.733 −5858 3401 88.7 157.1 35.1 9 121 (4) Y m55 10265.217 2.223 m53 10616.721 3.267 −5731 4501 214.3 −80.9 19.1 9 114 (5) Y m54 10332.327 3.635 m54 10364.062 5.446 −4492 5259 162.7 −261.5 59.2 9 119 (6) Y m54 10423.027 2.692 m54 10347.965 5.393 −3912 5346 42.7 −91.5 26.3 9 127 (7) Y m52 10725.185 3.640 m54 10353.804 5.393 −3791 4910 93.5 −141.7 38.8 9 137 (8) Y m54 10395.794 2.176 m53 10532.234 3.929 −10723 5853 438.5 −624.0 101.2 9 144 (9) Y m52 10754.753 2.832 m53 10555.649 5.446 −1966 4636 34.7 −95.5 40.6 7 74 (10) Y m52 10818.274 3.960 m52 10849.465 5.539 −6724 5848 720.5 −1601.5 405.4 9 84 (11) J m46 12267.888 3.274 m44 12615.928 4.638 −3025 2980 445.8 1400.6 482.9 6 123 (12) J m45 12556.996 2.279 m44 12648.741 4.607 −6714 4629 214.2 −82.8 16.6 9 134

(1) 10081.393 / 9868.186 (2) 10742.550 / 10065.045 (3) 10423.743 / 10216.313 (4) 10265.217 / 10616.721 5500 5500 5500 5500

5000 5000 5000 5000

4500 4500 4500 4500

4000 4000 4000 4000

3500 3500 3500 3500 0.7 0.5 0.3 0.1 0.1 1.2 1.0 0.8 0.6 0.4 0.6 0.4 0.2 0.0 0.2 0.4 0.2 0.0 0.2 0.4 (5) 10332.327 / 10364.062 (6) 10423.027 / 10347.965 (7) 10725.185 / 10353.804 (8) 10395.794 / 10532.234 5500 5500 5500 5500

5000 5000 5000 5000 ]

K 4500 4500 4500 4500 [

f f

e 4000 4000 4000 4000 T 3500 3500 3500 3500 0.2 0.0 0.2 0.4 0.6 0.2 0.0 0.2 0.4 0.6 0.3 0.1 0.1 0.3 0.5 0.2 0.0 0.2 0.4 0.6 (9) 10754.753 / 10555.649 (10) 10818.274 / 10849.465 (11) 12267.888 / 12615.928 (12) 12556.996 / 12648.741 5500 5500 5500 5500

5000 5000 5000 5000

4500 4500 4500 4500

4000 4000 4000 4000

3500 3500 3500 3500 0.2 0.0 0.2 0.4 0.6 0.2 0.0 0.2 0.4 0.6 0.8 0.6 0.4 0.2 0.0 0.4 0.2 0.0 0.2 0.4

log rj

Figure 4. LDR–Teff relations of the twelve Fe i–Fe i line pairs that we selected (also see Table 6). The line-pair ID together with the wavelengths (A)˚ of low- and high- excitation lines are indicated at the top of each panel. Blue data points show the relation between the observed log(LDR) and effective temperatures in literature for red giants. Red lines indicate the best-fitting relations that we obtained, Teff = a log(LDR) + b, with light-red shaded areas for the 1σ confidence intervals.

agated from the errors in the LDRs and coefficients of the −1/2 relations. Npair  Eq9 X 1 In order to examine the dependence of the relations on ∆T =   . (9) LDR ∆T 2 T and [Fe/H], we determined T of the group of the nine j=1 j eff LDR red giants (Table 2). Table 7 summarizes the re-determined Eq8 The former (∆TLDR) is the weighted standard error of TLDR effective temperatures and their standard errors in this work, Eq9 used by Paper I, and the latter (∆TLDR) is the error prop- together with those estimated on the basis of the relations

MNRAS 000,1–19 (2020) 12 D. Taniguchi et al.

Eq9 Table 7. Effective temperatures of nine red giants in kelvin. Teff is the literature effective temperatures adopted in this work. ∆TLDR in Eq8 group (T18) is calculated in this work, whereas TLDR and ∆TLDR in group (T18) are adopted from Paper I, in which they calculated the values in the same way as in this work.

Paper I (T18) This work (TW) Eq8 Eq9 Eq8 Eq9 Object Teff TLDR ∆TLDR ∆TLDR Npair TLDR ∆TLDR ∆TLDR Npair ε Leo 5398 ± 31 5429 24 17 42 5420 93 60 9 κ Gem 5029 ± 47 4982 13 10 60 4953 14 32 10 ε Vir 4983 ± 61 4996 11 10 65 4999 26 29 9 Pollux 4858 ± 60 4829 12 11 73 4829 50 46 11 µ Leo 4470 ± 40 4434 15 7 80 4517 38 26 12 Alphard 4171 ± 52 4143 9 7 79 4134 7 26 12 Aldebaran 3882 ± 19 3887 6 7 78 3905 27 42 12 α Cet 3796 ± 65 3780 9 6 79 3737 25 32 12 δ Oph 3783 ± 20 3812 4 6 79 3833 15 29 12

100 100

50 50 ] ] K K [ [

0 0 f f f f e e T T R R D D

L 50 L 50 T T

100 100

150 150 3750 4000 4250 4500 4750 5000 5250 5500 0.6 0.4 0.2 0.0 0.2 0.4 Teff [K] [Fe/H] [dex]

Figure 5. Differences between the re-determined and literature effective temperatures (TLDR(TW) and Teff , respectively) for red giants are plotted against the literature stellar parameters (Teff and [Fe/H]). Red lines show the linear regression lines of blue points with the light red shades showing the 1σ confidence intervals.

Eq8 in Paper I. In the table and hereafter, the former and lat- line pairs, and thus we simply adopted ∆TLDR(T18) as in ter cases are abbreviated as ‘TW’ (as of This Work) and Paper I. ‘T18’, respectively, and the parameters based on either of Figure 5 demonstrates the differences between the re- them are distinguished with the corresponding abbreviated determined effective temperatures TLDR(TW) and those in Eq8 word in parentheses as the suffix; e.g., ∆TLDR(TW) denotes literature Teff . It shows no apparent correlation between the the error of the LDR effective temperature on the basis of difference and either of Teff and [Fe/H]. Our TLDR was con- Equation 8 calculated with the relations in this work (TW). sistent with that in literature within ∼ 50 K over the entire To estimate the standard errors according to Equation 8 range of Teff and [Fe/H] of our sample red giants for the cal- Eq9 is, however, problematic in the case of the relations in this ibration. Besides, the error bars on the basis of ∆LDR(TW) Eq8 work for two reasons; (1) the error of ∆TLDR(TW) may be and ∆Teff explain the scatter around zero well, indicating large because the number of the line pairs is small, and that the error estimates were reasonable. Eq8 (2) the dispersion of Tj , and hence ∆TLDR(TW), may be underestimated because the nine red giants themselves were used to calibrate the LDR–Teff relations. The errors accord- 4 DETERMINATION OF EFFECTIVE ing to Equation 9 are more likely to be closer to the true TEMPERATURES OF RED SUPERGIANTS values as long as the errors of the LDRs are accurately Eq8 4.1 LDR temperatures of red supergiants estimated. In fact, ∆TLDR(TW) tends to be significantly smaller than ∆T Eq9 (TW) probably because of the afore- LDR Our new LDR–Teff relations for Fe i–Fe i line pairs are sup- mentioned reasons. Therefore, we concluded that the latter, posed to be less affected by the surface gravity and line Eq9 Eq8 ∆TLDR(TW), is more robust and reliable than ∆TLDR(TW). broadening than the relations by Paper I as a result of our Eq8 In contrast, the error ∆TLDR(T18) was found to be similar careful selection of the isolated Fe i lines. We here apply the Eq9 to ∆TLDR(T18) probably because of the large number of relations directly to ten nearby RSGs in the same way as to

MNRAS 000,1–19 (2020) Teff of RSGs using LDRs of Fe i lines in YJ bands 13

Table 8. Effective temperatures of ten RSGs using different sets of LDR–Teff relations in kelvin. TTiO(L05) shows the effective temperatures estimated with the optical TiO bands by Levesque et al.(2005) for comparison.

Paper I (T18) This work (TW) Eq8 Eq9 Eq8 Eq9 Object TTiO(L05) TLDR ∆TLDR ∆TLDR Npair TLDR ∆TLDR ∆TLDR Npair ζ Cep 4000 4340 22 13 72 4077 35 53 11 41 Gem 3900 4021 16 9 81 3944 29 45 12 ξ Cyg 3800 3989 15 10 81 3894 25 41 12 V809 Cas 3750 3960 18 10 77 3772 36 47 12 V424 Lac 3800 3934 19 13 78 3755 49 64 12 ψ1 Aur 3750 4049 24 10 75 3742 65 47 12 TV Gem 3700 4008 23 13 69 3681 109 66 9 BU Gem 3800 3970 26 14 71 3883 62 73 10 Betelgeuse 3650 3862 20 13 75 3618 40 66 12 NO Aur 3700 3849 19 11 77 3659 34 58 12

Cep 41 Gem Cyg V809 Cas V424 Lac 5500 5500 5500 5500 5500

] Taniguchi+18

K 5000 5000 5000 5000 5000 This Work (Fe-Fe) [

n 4500 4500 4500 4500 4500 o i

t 4000 4000 4000 4000 4000 a l 3500 3500 3500 3500 3500 e r 3000 3000 3000 3000 3000

R 3 2 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 2

D 1 Aur

L TV Gem BU Gem Betelgeuse NO Aur

5500 5500 5500 5500 5500 h

c 5000 5000 5000 5000 5000 a e

4500 4500 4500 4500 4500

m 4000 4000 4000 4000 4000 o r f

3500 3500 3500 3500 3500 f f e 3000 3000 3000 3000 3000

T 3 2 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 2 [eV]

Figure 6. Effective temperatures derived with the LDR relations (blue) in Paper I and (orange) of this work for RSGs are plotted against the difference in the ionization energies of the two lines in each line pair. The weighted mean values TLDR of the data points from the two sets of the relations are indicated by horizontal dashed lines in the respective colours, shaded with the width corresponding to the Eq8 standard errors ∆TLDR (Equation 8).

red giants in Section 3.4. Table 8 lists the resultant TLDR rors. Moreover, the residuals from the ∆χ–Tj correlation for of our targets together with the temperatures derived using most of the line pairs can be explained with the measure- the relations of Paper I. ment error. Therefore, the combination of the ∆χ effect and the measurement error may have a larger impact on T esti- The accuracy of our T of RSGs strongly depends j LDR mation for most line pairs with multiple species than other on the assumption that empirical LDR–Teff relations of Fe i effects (e.g. 3D non-LTE correction). lines are well consistent between giants and supergiants. Pa- Eq9 per II concluded that, at Teff = 4500 and 5000 K, the LDRs Concerning the error in TLDR, ∆TLDR(T18) is found to Eq8 of Fe i lines of dwarfs, giants and supergiants agree with each be systematically (1.5–2 times) smaller than ∆TLDR(T18); other. In order to test whether the agreement is also found this fact suggests that Tj for T18, hence TLDR(T18), has sys- between red giants and RSGs with 3500 . Teff . 4000 K, tematic errors introduced by the effect of the surface grav- Eq9 we first compare TLDR(TW) and TLDR(T18) to examine the ity. In contrast, ∆TLDR(TW) is similar to or larger than Eq8 dependence of the LDRs of multiple species on the surface ∆TLDR(TW). This fact implies that the scatter of Tj of gravity. The effect of the surface gravity on the LDRs is, Fe i line pairs with each star can be explained mostly with at least to some extent, governed by the difference in the the expected statistical errors except for a few cases; three ionization energies, ∆χ, of the elements forming low- and stars (TV Gem, BU Gem and ψ1 Aur) have larger line broad- high-excitation potential lines (Paper II). In fact, a weak ening than the other RSGs, which could slightly bias their anti-correlation of ∼ −80 K/eV between ∆χ and Teff from TLDR. However, the effect could be negligible because the individual LDR relations (Tj ) is seen for the ten RSGs (Fig- lines that we used are well isolated ones. We also examine ure 6). This trend is similar to the one observed by Paper II the error budget of TLDR(TW) using Tj − TLDR, which is a for giants and supergiants at higher temperatures. It indi- measure of the error in Tj (blue dots in Figure 7). Though cates that the surface gravity effect on the LDRs remains Tj − TLDR with each Fe i line pair has a distribution with similar down to the low-temperature ranges of the RSGs a non-zero offset, the offsets are within ∼ ±150 K for the and, therefore, TLDR(T18) of the RSGs have systematic er- pairs except the pair ID (2), which has an unpredictable

MNRAS 000,1–19 (2020) 14 D. Taniguchi et al. large offset (∼ 500 K). These offsets are smaller than the 1000 measurement error, which indicates, again, that the error MOOG (LTE) in TLDR(TW) is dominated by the expected statistical er- Bergemann (LTE) ror. Moreover, the small systematic offsets may be partly Bergemann (non-LTE) ] 500 K explained by other lines contaminating the Fe i lines that we [

Observed

j Tj TLDR T use. n o

Consequently, our new LDR–Teff relations with only s a i

b 0 iron lines (∆χ = 0 eV) are not affected by the surface grav- c

Eq9 i t ity. Since ∆TLDR may be slightly overestimated (see the dis- a m e

cussion on the ‘effective’ S/N in Section 2.3.2), we adopt t Eq8 s y

∆TLDR as the more accurate error of TLDR for the RSGs S 500 Eq9 Eq9 than ∆TLDR. Note that since ∆TLDR is calculated using the scatter of Tj , this error includes a part of the systematic error on Tj . The inclusion of the systematic error and the larger 1000 Eq9 1 2 3 4 5 6 7 8 9 10 11 12 measurement error may explain the larger ∆TLDR(TW) of RSGs than that of cool red giants. LDR ID In order to further examine the consistency between the LDR–Teff relations of red giants and RSGs, we compare the Fe i LDRs of the two groups by making use of their model Figure 7. Blue dots show the empirical deviation between the tem- spectra synthesized with MOOG (see Section 3 for spectral peratures from a given line pair, Tj , and the LDR temperatures, T (TW), for each Fe i line pair for each star. Orange squares, synthesis). We calculated the differences ∆ log rj in the log- LDR green triangles and red inverted triangles show theoretically- arithm of the LDRs log rj between RSG3 and RGB1, which estimated systematic bias (aj ∆ log rj ) of the LDR-based temper- have the same Teff of 3850 K (see Table 3 for their stellar atures Tj caused by the difference between red giants and RSGs parameters). Though ∆ log rj calculated with the imperfect calculated with MOOG, Bergemann’s tool with LTE and that model spectra cannot reproduce observed one in a quanti- with non-LTE, respectively. tative sense, aj ∆ log rj can be treated as a measure of the systematic error on Tj due to the difference in the LDR– Teff relations between red giants and RSGs. Figure 7 shows dots in Figure 7 show consistent trends for the pairs ID (4) aj ∆ log rj and Tj − TLDR for each line pair. Except the pair and (7), we found conflicting results for the pairs ID (2) and ID (2), aj ∆ log rj calculated with three types of synthetic (3). The former two line pairs with the small non-LTE effect spectra and observed Tj − TLDR are similar to each other indicate that the systematic bias caused by the non-LTE in a qualitative sense. Moreover, since aj ∆ log rj calculated effect is ∼ 100 K or less, though we cannot draw a strong with MOOG has a nearly symmetric distribution around conclusion based on the two pairs only. Moreover, it is hard the zero, the systematic errors for individual line pairs are to predict the systematic bias caused by the non-LTE ef- expected to be cancelled out by taking an average of Tj . fects without the non-LTE calculations for all the lines fully Though the reason of the inconsistency of the pair ID (2) is performed. We note that the 3D effect could also have an unclear, it might be safe to reject this pair because this pair impact on line depths of late-type stars (e.g. Collet et al. is suggested to involve a large (& 200 K) systematic error 2007; Kuˇcinskas et al. 2013), but the lack of a comprehen- both observationally and theoretically. We, thus, regard the sive grid of 3D model atmospheres of RSGs prevents us from weighted mean of Tj after excluding the pair ID (2) as the estimating its impact. final estimate of the LDR temperature of RSGs (Table 9). Since some lines of red giants and RSGs suffer from the non-LTE effects in the extended atmospheres of these 4.2 Comparison with previous Teff determinations objects (e.g. Bergemann et al. 2012b; Lind et al. 2012), we Betelgeuse is one of the most studied RSGs, whose Teff is finally examine how much the non-LTE effects could affect suggested to be free from strong time variability (White the LDR temperatures. For this purpose, we used the on- & Wing 1978; Bester et al. 1996; Gray 2008b; Levesque & line spectral synthesis tool developed by M. Bergemann’s Massey 2020), and thus it must be a good standard RSG for 4 group (Kovalev et al. 2019) to synthesize model spectra comparing Teff by different methods. Many previous works with and without the non-LTE effect. This tool can account have estimated Teff of Betelgeuse using a variety of meth- for the non-LTE effect for many Fe i lines calculated by ods, e.g., broad-band interferometry (e.g. Dyck et al. 1998; Bergemann et al.(2012a). However, the line list of the tool Haubois et al. 2009), SED fitting (e.g. Tsuji 1976; Scar- contains only half of the Fe i lines that we have selected, gle & Strecker 1979), the TiO method (e.g. Levesque et al. and we could calculate the LDRs of only four of the final 2005; Levesque & Massey 2020) and the excitation balance line pairs with the tool (IDs 2, 3, 4 and 7). The difference in of CO molecular lines (e.g. Lambert et al. 1984; Carr et al. aj ∆ log rj between LTE and non-LTE model spectra gives 2000), as summarized in Dolan et al.(2016). These methods an estimate of the systematic bias of the LDR temperature may, however, be affected by the systematic uncertainties caused by the non-LTE effect: ∼ 200 K for ID (2), ∼ 250 K described in Section 1.1. Davies et al.(2010) used the J - for ID (3), ∼ 0 K for ID (4) and ∼ 100 K for ID (7) (see band technique, which they claim is less affected than some Figure 7). While the observational results indicated by blue other methods by these systematic uncertainties, and ob- tained 3520±160 and 3660±170 K for spectra with spectral resolutions of R = 2, 000 and 6, 000, respectively. Our result 4 Last accessed on 2020 June 1. of 3611 ± 38 K for Betelgeuse is consistent with theirs. Their

MNRAS 000,1–19 (2020) Teff of RSGs using LDRs of Fe i lines in YJ bands 15

Table 9. Final estimates of effective temperatures and luminosities of ten RSGs derived with 11 LDR–Teff relations except ID (2) (see Section 4.1). The statistical error associated with TLDR is calculated with Equation 8. Fourth and fifth columns show the parallax tabulated in the Gaia EDR3 catalogue (Gaia Collaboration et al. 2020) and the bias involved in it estimated following Lindegren et al. (2020a), respectively, except Betelgeuse. Its parallax marked with an asterisk is from the Hipparcos catalogue (van Leeuwen 2007).

Object HD TLDR[K] Npair Parallax [mas] Bias [mas] log(L/L ) +0.07 ζ Cep 210745 4073 ± 31 10 3.297 ± 0.146 −0.022 3.57−0.07 +0.33 41 Gem 52005 3940 ± 29 11 0.721 ± 0.091 −0.033 4.28−0.27 +0.09 ξ Cyg 200905 3891 ± 24 11 2.836 ± 0.127 −0.023 3.71−0.09 +0.11 V809 Cas 219978 3768 ± 35 11 0.999 ± 0.039 −0.032 4.30−0.11 +0.11 V424 Lac 216946 3749 ± 49 11 1.402 ± 0.113 −0.027 4.06−0.10 1 +0.27 ψ Aur 44537 3740 ± 67 11 0.443 ± 0.110 −0.036 4.94−0.22 +0.39 TV Gem 42475 3676 ± 117 8 0.478 ± 0.135 −0.029 5.41−0.28 +0.43 BU Gem 42543 3883 ± 67 9 0.564 ± 0.125 −0.043 5.10−0.30 ∗ +0.14 Betelgeuse 39801 3611 ± 38 11 6.55 ± 0.83 4.91−0.13 +0.27 NO Aur 37536 3651 ± 31 11 0.919 ± 0.093 −0.042 4.92−0.22 estimate for V424 Lac, 3580 ± 230 K, based on its spectrum with R = 2, 000 is also consistent with ours, 3749 ± 49 K, within the errors, but the variability of V424 Lac is poorly 4100 known. We note that Teff of no other RSGs in our sam- ple were measured by Davies et al.(2010) and other recent 4000 works using the J -band technique. ] K [ 3900 )

Another promising approach is the spectro- 5 0 L interferometric observation, which is less affected by (

O the outer envelopes of RSGs. Ohnaka et al.(2011) de- i 3800 T T termined Betelgeuse’s Teff to be 3690 ± 54 K with this approach. In addition Arroyo-Torres et al.(2013) estimated 3700 it to be 3620 ± 137 K, using the angular diameter measured by Ohnaka et al.(2011). Our result of 3611 ± 38 K for Betelgeuse is consistent with theirs, and hence supports the 3600 validity of the Teff scale that we obtained. Unfortunately, no other RSGs than Betelgeuse in our sample have the 3600 3700 3800 3900 4000 4100 Eq8 (TW) [K] spectro-interferometric Teff in literature to compare with. TLDR ± TLDR

Figure 8 compares the effective temperatures of all the Eq8 Figure 8. Comparison between the effective temperatures of RSGs RSGs in this work (TLDR ±∆TLDR) and those determined by determined in this work (horizontal axis) and those by Levesque Levesque et al.(2005) ( TTiO(L05)), where the TiO method et al.(2005) (vertical axis). A red line shows the linear regres- was employed. Whereas comparing Teff of each star is not so practical, given that the time variations of T for RSGs are sion line of blue points with the light red shade showing the 1σ eff confidence intervals. as large as several hundreds kelvins in the worst cases (e.g. Levesque et al. 2007; Clark et al. 2010; Wasatonic et al. 2015), the comparison of these as a whole tells something 4.3 Red supergiants on the HR diagram significant; the two sets of Teff were found to be well con- Eq8 sistent, considering the errors of the former (∆TLDR) and In many previous works, more than a decade ago, latter (∼ 50 K), but with a slope of 0.70 ± 0.14, which is observationally-determined Teff of the brightest slightly different from one. This consistency indicates that RSGs (Mbol < −7 mag) were often lower than expected from the TiO method yields not strongly biased Teff of RSGs that theoretical models at a given luminosity (see, e.g., Figure 8 have the solar abundance pattern. However, it is unclear in Massey 2003). Since then, several observational studies whether the TiO method can yield unbiased Teff of RSGs have compared their Teff with theoretical models, mainly that have non-solar abundances, considering the uncertain- with Geneva’s stellar evolution model (Ekstr¨om et al. ties discussed in Section 1.1. In fact, Levesque et al.(2006) 2012; Georgy et al. 2013), on the Hertzsprung-Russel (HR) and Davies et al.(2013) determined Teff of common ∼ 20 diagram. Some of them found good consistency between RSGs in the Magellanic Clouds, using the TiO method, but them (e.g. Levesque et al. 2005; Davies et al. 2013; Wit- Teff by Levesque et al.(2006) are systematically ∼ 100 K tkowski et al. 2017), whereas some others found offsets higher than those by Davies et al.(2013). Moreover, Davies between the measurements and theoretical models (e.g. et al.(2013) compared Teff derived with the TiO method Levesque et al. 2006; Tabernero et al. 2018). to that with the SED method, and found that the latter In order to plot our observed RSGs on the HR diagram, method yielded ∼ 400 K higher Teff than the former. we derived the bolometric luminosity, using the parallax and

MNRAS 000,1–19 (2020) 16 D. Taniguchi et al. bolometric magnitude of each source as follows. As for the parallaxes, we used the values in Gaia EDR3 (Gaia Col- 32 5.5 Minit = MSun 30 laboration et al. 2016, 2020) for all our RSG samples except Minit = 25MSun

Betelgeuse, for which we used the parallax in the Hipparcos Minit = 20MSun 5.0 catalogue by van Leeuwen(2007) because Gaia EDR3 has 25 Minit = 15MSun ] n no entry for it. Since there is a systematic bias in the Gaia ) u n S

u 4.5 S parallax (Lindegren et al. 2020b), we followed the recipe by 20 M L [ /

Minit = 12MSun t L n ( Lindegren et al.(2020a) to reduce the bias. In brief, we in- e r r g 4.0 u o c putted G-band magnitude, effective wavenumber and eclip- l Minit = 09MSun 15 tic latitude tabulated in Gaia EDR3 into the code developed M 5 3.5 by them to calculate the bias estimate. Estimated biases, Minit = 07MSun summarized in Table 9 and ranging from −43 to −22 as, 10 µ 3.0 only change the final log L by up to 0.07, which is smaller Minit = 05MSun This Work than the statistical error in it. Nevertheless, it should be kept 5 in mind that (i) all the sample RSGs have G < 6 mag, which 5500 5000 4500 4000 3500 [K] is out of the range of the bias calibration by Lindegren et al. Teff (2020a), and (ii) the Gaia parallaxes of very bright stars could contain large systematic biases; for example, several stars with G . 4 mag and with parallaxes larger than 5 mas Figure 9. HR diagram of RSGs. Blue circles show the effective have systematic biases of & 1 mas in Gaia DR2 (Drimmel temperatures and bolometric luminosities of the sample RSGs et al. 2019) though the bias would be smaller in EDR3 (Torra determined in this work. The solar-metallicity Geneva evolution- et al. 2020), and Gaia DR3 parallaxes of all our sample RSGs ary track by Ekstr¨om et al.(2012) with and without rotation are shown with dotted and solid lines, respectively, colour-coded with are smaller than 4 mas. We note that the statistical errors of the current mass. the parallaxes of our targets tend to be larger than those ex- pected from their magnitudes in both the Hipparcos (van Leeuwen 2007) and Gaia EDR3 (Lindegren et al. 2020b) tribution of the data points of our sample RSGs in our esti- catalogues. It is most likely, at least partly, due to strong mates with those expected from the latest Geneva’s stellar granulation in RSGs, which fluctuates the positions of their evolution model with the solar metallicity (Ekstr¨om et al. photometric centroids by ∼ 0.1 AU (Chiavassa et al. 2011a; 2012). Although the sample size is limited, Teff obtained in Pasquato et al. 2011). this work are consistent with the range of Teff in which RSGs As for the bolometric magnitude, we estimated it are expected to stay relatively long time. A larger sample of for each RSG sample from the 2MASS K s-band magni- RSGs, especially those with a higher luminosity, would en- tude (Skrutskie et al. 2006), considering the extinction and able us to investigate the properties of the Galactic RSGs, the bolometric correction. The sampled epochs of the cata- e.g., comparing the Teff distribution more closely with var- logued K s-band magnitude that we used are different from ious evolutionary models, determining the relation between those of their WINERED observations. However, the dif- the spectral type and Teff and so on. ference in the epochs would not be a significant problem, given that the magnitude variation of the RSG in the in- frared bands is known to be negligible (e.g. Yang et al. 2018; Ren et al. 2019), in contrast to the optical variation, 5 SUMMARY AND FUTURE PROSPECTS which is as large as a few magnitudes (Kiss et al. 2006; In this paper, we calibrated the empirical relations between Soraisam et al. 2018, and references therein). Concerning twelve LDRs of Fe i lines and Teff , using nine solar-metal red the interstellar and circumstellar extinction, we converted giants observed with WINERED. These relations enabled us the V -band extinction A(V ) of the RSGs measured by to determine Teff of red giants to a precision of ∼ 30 K in the Levesque et al.(2005), with the typical precision of 0 .15 mag, best cases, i.e., early-M type giants with good S/N. We ap- to the K s-band extinction A(Ks), using the reddening law plied these relations to ten nearby RSGs and obtained Teff to A(Ks)/A(V ) = 0.116 by Cardelli et al.(1989), assuming the a precision of ∼ 40 K. The estimated Teff are in good agree- total-to-selective extinction ratio RV = 3.1 as in Levesque ment with the values estimated with different methods in et al.(2005). We estimated the bolometric correction for the relevant literature: those with the TiO method by Levesque K s band, BCKs , interpolating the BCK value with regard et al.(2005) and that of Betelgeuse on the basis of spectro- to the obtained Teff (Section 4.1) in the tabulated relation interferometry by Ohnaka et al.(2011) and Arroyo-Torres between Teff and BCK for RSGs in Table 6 of Levesque et al. et al.(2013), as well as those expected in Geneva’s stellar (2005). Since the extinction to our sample RSGs are small, evolution model (Ekstr¨om et al. 2012). A(V ) = 2.17 mag at most, the choice of the reddening law Our method uses only Fe i lines in the near-infrared (the and the RV value and the difference between K s and K do Y and J bands) high-resolution spectra. Because of it, our not significantly affect the resultant luminosities. Finally, the method is expected to give more unbiased Teff of RSGs than bolometric luminosity (L) of each target RSG and its error the other published spectroscopic methods, which rely on were calculated using the Monte Carlo method (Table 9). lines of molecules and/or multiple atoms and inevitably suf- In the HR diagram in Figure 9, we compared the dis- fer from some significant systematic uncertainties. First, our method is independent of chemical abundance ratios be- cause our method relies on only one species, Fe i. Second, 5 https://gitlab.com/icc-ub/public/gaiadr3_zeropoint our method is expected to be less affected by the granulation

MNRAS 000,1–19 (2020) Teff of RSGs using LDRs of Fe i lines in YJ bands 17 than the methods using molecular bands because the granu- agreement between the JSPS and the Department of Science lation tends to vary the temperature structure of the upper and Technology (DST) in India. KF acknowledges financial atmospheric layers in particular and, hence, the strengths of support of KAKENHI No. 16H07323. HS acknowledges fi- molecular bands. Finally, our method is less affected by the nancial support of KAKENHI No. 19K03917. surface gravity effect from which conventional LDR methods This work has made use of the VALD database, op- with multiple species suffer. erated at Uppsala University, the Institute of Astronomy Though there are possible systematic biases of the tem- RAS in Moscow, and the University of Vienna. This research peratures derived with individual line pairs due to the sur- has made use of the SIMBAD database, operated at CDS, face gravity, microturbulent velocity and/or line broadening Strasbourg, France. This work has made use of data from effects, it is expected that they cancel each other out when the European Space Agency (ESA) mission Gaia (https: the temperatures derived with several pairs are averaged. In //www.cosmos.esa.int/gaia), processed by the Gaia Data contrast, it is unclear whether the systematic bias due to Processing and Analysis Consortium (DPAC, https://www. the non-LTE effect, as large as ∼ 250 K for each pair, can be cosmos.esa.int/web/gaia/dpac/consortium). Funding for reduced by taking the average of individual estimates. Nev- the DPAC has been provided by national institutions, in par- ertheless, the final LDR temperatures using all the 11 pairs ticular the institutions participating in the Gaia Multilat- are well consistent within ∼ 100 K with the temperatures eral Agreement. This publication makes use of data products derived with the LDR ID (4), which is insensitive to the from the Two Micron All Sky Survey, which is a joint project non-LTE effect. This fact indicates that the actual system- of the University of Massachusetts and the Infrared Process- atic bias due to the non-LTE effect is as small as ∼ 100 K. ing and Analysis Center/California Institute of Technology, To further examine the possible systematic errors, in-depth funded by the National Aeronautics and Space Administra- consideration with reliable 3D non-LTE model spectra is de- tion and the National Science Foundation. sired. Our sample is limited to solar-metal objects, red giants and RSGs, and the relations we obtained in this work can DATA AVAILABILITY be applied to solar-metal objects only. In order to get rid of this limitation, the metallicity dependence of the LDRs in The data underlying this article will be shared on reasonable the Y and J bands needs to be examined. Observations with request to the corresponding author. recent and/or future near-infrared high-resolution spectro- graphs with high sensitivities (e.g. WINERED; Ikeda et al. 2016, 2018) will provide sufficient-statistics data of RSGs at large distances, e.g., those in the inner/outer-Galaxy and REFERENCES some local-group galaxies like the Magellanic Clouds and Alonso-Santiago J., Negueruela I., Marco A., Tabernero H. M., more. 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Yang M., et al., 2018, A&A, 616, A175 van Leeuwen F., 2007, A&A, 474, 653

This paper has been typeset from a TEX/LATEX file prepared by the author.

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